Optimal. Leaf size=271 \[ \frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{5/2}}{5 a \sqrt{a^2 x^2+1}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}-\frac{3 a x^2 \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{8 \sqrt{a^2 x^2+1}}-\frac{3 \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{16 a \sqrt{a^2 x^2+1}} \]
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Rubi [A] time = 0.272977, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {5682, 5675, 5663, 5779, 3312, 3307, 2180, 2204, 2205} \[ \frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{5/2}}{5 a \sqrt{a^2 x^2+1}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}-\frac{3 a x^2 \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{8 \sqrt{a^2 x^2+1}}-\frac{3 \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{16 a \sqrt{a^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5682
Rule 5675
Rule 5663
Rule 5779
Rule 3312
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2} \, dx &=\frac{1}{2} x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac{\sqrt{c+a^2 c x^2} \int \frac{\sinh ^{-1}(a x)^{3/2}}{\sqrt{1+a^2 x^2}} \, dx}{2 \sqrt{1+a^2 x^2}}-\frac{\left (3 a \sqrt{c+a^2 c x^2}\right ) \int x \sqrt{\sinh ^{-1}(a x)} \, dx}{4 \sqrt{1+a^2 x^2}}\\ &=-\frac{3 a x^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{8 \sqrt{1+a^2 x^2}}+\frac{1}{2} x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{5 a \sqrt{1+a^2 x^2}}+\frac{\left (3 a^2 \sqrt{c+a^2 c x^2}\right ) \int \frac{x^2}{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}} \, dx}{16 \sqrt{1+a^2 x^2}}\\ &=-\frac{3 a x^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{8 \sqrt{1+a^2 x^2}}+\frac{1}{2} x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{5 a \sqrt{1+a^2 x^2}}+\frac{\left (3 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a \sqrt{1+a^2 x^2}}\\ &=-\frac{3 a x^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{8 \sqrt{1+a^2 x^2}}+\frac{1}{2} x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{5 a \sqrt{1+a^2 x^2}}-\frac{\left (3 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}-\frac{\cosh (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{16 a \sqrt{1+a^2 x^2}}\\ &=-\frac{3 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{16 a \sqrt{1+a^2 x^2}}-\frac{3 a x^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{8 \sqrt{1+a^2 x^2}}+\frac{1}{2} x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{5 a \sqrt{1+a^2 x^2}}+\frac{\left (3 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a \sqrt{1+a^2 x^2}}\\ &=-\frac{3 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{16 a \sqrt{1+a^2 x^2}}-\frac{3 a x^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{8 \sqrt{1+a^2 x^2}}+\frac{1}{2} x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{5 a \sqrt{1+a^2 x^2}}+\frac{\left (3 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt{1+a^2 x^2}}+\frac{\left (3 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt{1+a^2 x^2}}\\ &=-\frac{3 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{16 a \sqrt{1+a^2 x^2}}-\frac{3 a x^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{8 \sqrt{1+a^2 x^2}}+\frac{1}{2} x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{5 a \sqrt{1+a^2 x^2}}+\frac{\left (3 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{32 a \sqrt{1+a^2 x^2}}+\frac{\left (3 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{32 a \sqrt{1+a^2 x^2}}\\ &=-\frac{3 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{16 a \sqrt{1+a^2 x^2}}-\frac{3 a x^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{8 \sqrt{1+a^2 x^2}}+\frac{1}{2} x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{5 a \sqrt{1+a^2 x^2}}+\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{c+a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{1+a^2 x^2}}+\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{c+a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{1+a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.271102, size = 126, normalized size = 0.46 \[ \frac{\sqrt{c \left (a^2 x^2+1\right )} \left (15 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )+15 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )+8 \sqrt{\sinh ^{-1}(a x)} \left (4 \sinh ^{-1}(a x) \left (4 \sinh ^{-1}(a x)+5 \sinh \left (2 \sinh ^{-1}(a x)\right )\right )-15 \cosh \left (2 \sinh ^{-1}(a x)\right )\right )\right )}{640 a \sqrt{a^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.218, size = 0, normalized size = 0. \begin{align*} \int \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}c{x}^{2}+c}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a^{2} c x^{2} + c} \operatorname{arsinh}\left (a x\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a^{2} c x^{2} + c} \operatorname{arsinh}\left (a x\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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